Abstract
The present paper proposes a new unconditionally stable method to solve telegraph equation by using associated Hermite (AH) orthogonal functions. Unlike other numerical approaches, the time variables in the given equation can be handled analytically by AH basis functions. By using the Galerkin’s method, one can eliminate the time variables from calculations, which results in a series of implicit equations. And the coefficients of results for all orders can then be obtained by the expanded equations and the numerical results can be reconstructed during the computing process. The precision and stability of the proposed method are proved by some examples, which show the numerical solution acquired is acceptable when compared with some existing methods.
1. Introduction
In this work, the following telegraph equation is considered:with the initial conditionsand the Dirichlet boundary conditionswhere and are real constants. We now assume that , and , are continuous functions of and , respectively. The telegraph equation has been arisen in the propagation of electrical signals in wave phenomena and transmission line.
During past years, much literatures have paid attention to the analysis and development of telegraph equation, see, for example, [1–4]. Mohanty et al. [5–7] developed the finite difference method for solving telegraph equation and it has been proved to be of high accuracy. Dehghan and Shokri [8] proposed a new scheme for telegraph equation using collocation points and radial basis function. In [2], a high-order accurate scheme is introduced for telegraph equation. Dehghan and Ghesmati [9] developed the boundary integral equation scheme for solving second-order hyperbolic equation. Saadatmandi and Dehghan [10] proposed the Chebyshev Tau method for telegraph equation. In [11], the Chebyshev cardinal function is used for solving the telegraph equation. L.-B. Liu and H.-W. Liu [12] proposed a numerical solution for telegraph equation by applying the trapezoidal formula and quartic spline. Authors of [13–16] developed the radial basis function scheme for solving the telegraph equation. In recent years, some numerical schemes were proposed to solve (1) based on B-spline collocation method [17–19], polynomial scaling functions [20, 21], shifted Gegenbauer pseudospectral method [22], and differential quadrature algorithm [23]. These schemes are conditionally stable.
To construct an unconditional stability method to solve telegraph equation, Mohanty et al. [24–26] developed three-level alternating direction implicit scheme to solve telegraph equation. Borhanifar and Abazari [27] proposed the parallel difference method, which is unconditionally stable, to solve telegraph equation. In [28], Mohanty presented a new unconditionally stable method to solve telegraph equation, where two parameters were introduced. Gao and Chi [29] presented two semidiscretion schemes for solving the hyperbolic equations. H.-W. Liu and L.-B. Liu [30] developed the spline difference method to solve problem. In [31], the compact difference scheme is proposed for solving the telegraph equation. Xie et al. [32] developed a compact difference and ADI scheme for telegraph equations with a fourth order in space. Some high-order methods, using the Padé approximation method [33] and cubic Hermite interpolation [34], were proposed for the telegraph equations.
In this study, we attempt to solve the telegraph equation using associated Hermite (AH) orthogonal functions. The Hermite functions, which were widely studied in the Hermite spectral method (HSM), are constructed by Hermite basis functions with a translated and scaled weighting function [35]. Kavian and Funaro [36] considered the use of HSM combined with variable transformation technique to solve the diffusion problems in unbounded domains. In [37, 38], Guo et al. developed the Hermite spectral method and Hermite pseudospectral methods (HPSM), and Alıcı [39] used the HPSM to solve the Schrödinger equation. To stabilize the methods, a time-dependent parameter [40–42] is introduced for traditional Hermite functions to construct a conditional stability method. Unfortunately, the Hermite orthogonal functions were not used to structure an unconditional stability method for telegraph equation.
The fundamental aim of the literature is eliminating of the time variables in telegraph equation and then construct a new unconditionally stable method, the accuracy of which is absolutely independent of temporal step. In our presented method, the time variables in (1) are handled by the AH basis functions at the first step. Then, the Galerkin’s method and the central difference method are introduced to this handled equation; a series of equations without time variables can be obtained. Finally, we can solve this telegraph equation by using the chasing method and the numerical solution could be reconstructed by using coefficients of AH functions.
2. Associated Hermite Functions
AH functions can be expressed as where the Hermite polynomials is
The Hermite polynomials satisfy the following recursive relationship [43]:Then we have the derivative relations of Hermite polynomials:
By introducing a time-translating parameter, AH functions could be transformed to a causal form:where ; and are the translating parameter and time-scaling parameter, respectively. A function with causal relationships can be expanded by choosing a suitable and :
Combining (6) and (8), we haveProperty (7) and the above formula lead to the derivation of :
Combining (9) and (11), we haveLet ; we can obtainLet ; we haveThen we can deduce the partial derivative of
For the second partial derivative of , we havewhereFrom the above equations, we can obtain the first and the second partial derivative of :
3. Description of the Method
3.1. Construction of Computing Matrix
The partial differential with respect to can be written as
Rewriting the telegraph equation (1) using (9), (18), and (19), we can obtain
In (20), we introduced a Galerkin procedure [44] in order to eliminate the time variable terms . By multiplying each side of (20) by time-dependent term , we can obtain the following equation by integrating over time from 0 to ∞:where
Discretize (21) using the central difference scheme in space; we have
From (23), one could conclude that the variable of the -order is connected with adjacent fields, from to . In (23), is moved to the left side of the equation and simplified towhereIn (24), is an unit matrix, and is a -tupple matrix of AH space for numerical results.
Rewriting (24) with a nested matrix, we havewhereIn (26), is a combination of numerical result with all orders and all points. represents source coefficients in the computational domain. [] is a banded sparse matrix with submatrix elements.
3.2. Method’s Treatment of Boundary and Initial Conditions
The application of presented method to telegraph equation based on a concealment condition that the initial valve of the equation should be 0; that is . For the nonhomogeneous initial condition, we haveFor the Dirichlet boundary conditions, there are a couple of ways to deal with it. The point on the boundary would be regarded as a point source which changes over time in the first approach. Then we can expand , , and using AH basis functions, respectively. However, this kind of method has the drawback of complicated procedure and a little loss in accuracy. In this work, the second approach has been adopted in which the Dirichlet boundary conditions are homogenized. We applyIntroducing (28) and (29) to (1)–(3), we can obtain a telegraph equation with the homogeneous boundary and initial conditions
Rewriting (22) and (26) we have
To solve (32), the lower-upper (LU) decomposition scheme is applied to decompose [] initially, and iteration method is used to obtain the numerical results. The LU decomposition of [] is handled only once when starting calculation process. Finally, one can obtain the numerical result from the expansion coefficients as
Contrary to the other numerical approach, this presented method has an implicit relationship in each variables, which can be reflected in the sparse matrix []. One should notice that (32) is independent of the temporal testing procedure . And the matrix , which does not contain the order , remains unchanged in the calculation process. Therefore, the temporal step does not affect the stability of this method any more, and then an unconditionally stable scheme is structured. In this presented method, the time step size is only applied to compute the AH coefficients of the source term due to (22) and (31), which has been done at the beginning of the computation. We can choose a relatively small value of time step to calculate (22) or (31) accurately and describe the process of propagation of electrical signals clearly, and this procedure does not increase the computation time.
4. Numerical Examples
In this part, the following four examples of the telegraph equation with exact solution have been solved by the proposed scheme. To measure the accuracy and versatility of this proposed scheme, the following , , and root-mean-square (RMS) errors are reported:
Example 1. We consider the telegraph equation:over a region , with initial conditionsand boundary conditions
The exact solution of this equation is . The results for different value of α and β obtained by using the proposed method are compared with those obtained by Mohanty [28], H.-W. Liu and L.-B. Liu [30] and L.-B. Liu and H.-W. Liu [12], which are presented in Tables 1 and 2. The graph of analytical and numerical result at = 0.25, = 0.5 and = 0.75 is shown in Figure 1.
Example 2. In this case, the telegraph equation (1) with = 4, = 2 over a region [] with the following boundary and initial conditions was considered:
We have . And the exact solution of this example is . The , , and RMS errors for = 0.5 and 1.0 are presented in Table 3 with = 0.02 and = 0.0001. The errors are compared with the numerical results acquired in Dehghan and Shokri [8], Sharifi and Rashidinia [18] and Mittal and Bhatia [19]. The RMS errors for = 3.0 with = 0.01 are shown in Table 4. The numerical solutions by this proposed method are compared with the results obtained by [9] and another unconditionally stable method in [31]. The space-time graph of numerical result up to = 2 is shown in Figure 2.
Example 3. In this case, we consider telegraph equation (1) with = 10, = 5 over region , and the following boundary and initial conditions:
In this problem, we have . And the analytical solution of the example is . The , , and RMS errors at = 0.2 and = 0.4 with = 0.001, = 0.001 are shown in Table 5. And also the numerical results are compared with Sharifi and Rashidinia [18] and Mittal and Bhatia [19]. The space-time graphs of numerical result with = 0.02 and = 0.01 up to = 0.5 and 1.0 are presented in Figure 3.
(a)
(b)
Example 4. We consider telegraph equation with α = 0.5 and β = 1 in the interval , with initial conditionsand boundary conditions
The analytical solution of this equation is . The CPU time and , , and RMS errors for = 1 and 5 are shown in Table 6 with = 0.01 and = 0.001. The numerical solution is compared with results of [8, 19]. It can be concluded that numerical solutions obtained by this presented scheme are good in comparison with [8, 19]. Moreover, our method has a higher efficiency than the other two methods when the computation time gets longer. The space-time graph of numerical result is shown in Figure 4.
5. Conclusion
In the literature, we have constructed a new unconditionally stable method to solve the telegraph equation. Since the time variables have been eliminated from computations, the convergence and precision of the presented method are independent of the temporal step. To demonstrate the stability and accuracy of this proposed method, four numerical examples are conducted and compared with the numerical results available in previous literatures. The comparison numerical results reveal the unconditional stability and high precision of the presented scheme in the current literature.
The paper has provided a new mentality for solving telegraph equation and this method is now extended to multidimensional cases and other kinds of partial differential equations in a future study.
Competing Interests
The authors declare that they have no competing interests regarding the publication of this paper.